'Snap during transform' not doing anything
The reason why it does not seem to work is that the mouse cursor is not hovering exactly over the vertex which is to snap.




Enable Snap during transform by clicking on the red Magnet symbol or by pressing Shift + Tab.
Choose Vertex as Snap Element
Enable Snap onto itself 

 

*If Snap during transform is disabled you can hold down Ctrl during Translation (G).*





Hover the Mouse cursor over the desired Vertex

Make sure you hover exactly over the Vertex otherwise it won't snap


A little circle will appear on the Vertex 




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Another possibility to achieve the desired result is to Merge the Vertices


Select the two Vertice by holding down Shift and click RMB
Press W and choose Merge
Choose 'Merge at Last' or 'Merge at First' depending on the selection order